On the stability of plane Poiseuille flow to finite-amplitude disturbances, considering the higher-order Landau coefficients

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Abstract
In this work a study has been made of the Stuart (1960)–Watson (1960) formalism as applied to plane Poiseuille flow. In particular, the higher-order Landau coefficients have been calculated for the Reynolds & Potter (1967) method and for the Watson (1960) method. The results have been used to study the convergence of the Stuart–Landau series. A convergence curve in the (α, R)-plane has been obtained by using suitable Domb–Sykes plots. In the region of poor convergence of the series, and also in a part of the divergent region of the series, it has been found that the Shanks (1955) method, using the em1 transformation, serves as a very effective way of finding the proper sum of the series, or of finding the proper antilimit of the series. The results for the velocity calculations at R = 5000 are in very good agreement with Herbert's (1977) Fourier-truncation method using N = 4. The Watson method and the Reynolds & Potter method have also been compared inthe subcritical and supercritical regions. It is found in the supercritical region that there is not much difference in the results by the ‘true problem’ of Watson and the ‘false problem’ of Reynolds & Potter when the respective series in both methods are summed by the Shanks method. This fact could possibly be capitalized upon in the subcritical region, where the Watson method is difficult to apply.